Interpretation of Potential-Probe Measurements in Two-Carrier Structures

Abstract

Theoretical arguments and experimental evidence show that the potential $\overline{\phi }$, measured by a high-impedance metal probe to the surface of a semiconductor or insulator in which the electron and hole concentrations are nearly equal, is given by $\overline{\phi }=\frac{({\mu }_n{\phi }_n+{\phi }_p{\mu }_p)}{({\mu }_n+{\mu }_p)}$, where ${\phi }_n$ and ${\phi }_p$ are the electron and hole quasi Fermi levels and ${\mu }_n$ and ${\mu }_p$ are the electron and hole mobilities. As a consequence, the carrier concentration $n$ is given exactly by $n=\left|\frac{J}{\left[e{\mu }_p\mathrm{}\right(b+1\mathrm{}\left)\mathrm{}\frac{d\overline{\phi }}{\mathrm{dx}}\right]}\right|$ where $J$ is the current density. This relation constitutes a powerful tool for measuring $n$. The experimental evidence was found from measurements on silicon $p-i-n$ structures forward biased into the double injection region where $n\approx p$. The observed potential drops $\Delta V$ at the $p-i$ and $n-i$ junctions were compared with the values of the carrier concentrations $n_1$ at the junctions. The theoretical relation between $\Delta V$ and $n_1$ depends strongly on the assumption that $\overline{\phi }$ is the quantity measured by the probe. Thus the fact that the experimental data agree with the above mentioned theoretical relation is a strong confirmation of the use of $\overline{\phi }$. Theoretical arguments suggest that the generalization to the case $n\ne p$ is given by $\overline{\phi }=\frac{(n{\mu }_n{\phi }_n+p{\mu }_p{\phi }_p)}{(n{\mu }_n+p{\mu }_p)}$.